# Can superdense coding be made more efficient?

Superdense encoding allows us to transmit the information of two classical bits using a single qubit with a pre-shared Bell state qubit pair. We can duplicate the construct to transmit $$2n$$ classical bits using $$n$$ qubits. My question is: can we do better by using fewer qubits by leveraging multiqubit entanglement?

Superdense coding applies $$I, X, Z, XZ$$ operator on a single qubit to get $$4$$ different outcomes. For $$n$$ qubits, if we can only apply these four operators on individual qubits, then $$4^n$$ combinations are possible. Given $$4^n=2^{2n}$$, one can only get information of $$2n$$ classical bits. I know it's a big "if". Is that assumption correct since Pauli group forms a basis for all operators?

Sorry I am a beginner in this subject and am not clear about many concepts yet. Also apologize if the answer is already embedded in some earlier thread. Thanks.